Armodue harmony

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[[toc]]
=**Armodue: basic elements of harmony**= 

This is a translation of an article by Luca Attanasio. Original page in italian: [[http://www.armodue.com/armonia.htm]]
//Note: This is a preliminary tranlsation. Parts that are still "under construction" are marked with "XXX".//


----

=Chapter 1: Two theses supporting the system= 

==The supremacy of the fifth and and the seventh harmonic in Armodue== 

The twelve note system that has been ruling for several centuries is based on the third harmonic of the overtone series, which forms a perfect twelfth (octave-reducible to a perfect fifth) with the first harmonic or fundamental. Also, the pythagorean tradion - the cicle of fifths - is based on the perfect fifth and hence on the same frequency ratio 3:2.

But, if in the twelve note system the pitch of the third harmonic - hence the perfect fourth and the perfect fifth - are almost perfectly respected (the tempered fourth and fifth differ only of the fiftieth part of a semitone from the natural fifth and fourth), this cannot be said about the odd harmonics (even harmonics are not counted because they are simply duplicates in the octave of odd harmonics) immediately above the third one: the fifth and seventh harmonic.

In the tempered system the fifth and the seventh harmonic appear as major third and minor seventh intervals formed to the fundamental (equating all intervals to their octaver-reduced form for simplicity), but the sizes of the tempered major third and minor seventh do not match the sizes of the respective natural intervals.

In Armodue, in contrast, intervals corresponding to those formed by the fifth and seventh harmonic are rendered with greater fidelity of intonation. In this sense, Armodue increases the consonance of the higher harmonics; in particular, it renders the pitch of the seventh harmonic at maximum naturalness.

For this reason, especially important in Armodue are the interval of five eka (corresponding to the interval ratio given by the fifth harmonic) and the interval of 13 eka (corresponding to the interval ratio subsisting with the seventh harmonic). The circle of fifths which is the base of the dodecatonic system is replaced in Armodue by the cycle of 5 eka and the cycle of 13 eka, emphasizing the priority of the fifth and the seventh harmonic.

==The triple mean of the double diagonal / side of the square== 

From a philosophical point of view, the system of twelve notes was justified in the past by this mathematical property: the arithmetic mean and the harmonic mean of the octave (interval ratio 2:1) correspond to the perfect fifth (ratio 3:2) and the perfect fourth (ratio 4:3), while the geometric mean divides the octave exatcly into two tritone intervals (ratio: square root of 2).

Analogously, the philosophical foundation of Armodue and esadecafonia can be shown by calculating the three means of frequency geometrically equivalent to the ratio between the double of the diagonal (square root of 2 multiplied by 2) and the side length (of measure: 1) of a square.

The arithmetic mean is exactly the interval of nine eka, the geometric mean exactly twelve eka and finally the harmonic mean exactly equal to fifteen eka.


=Chapter 2: The interval table= 

==Qualitative categories of intervals== 

Armodue consists of sixteen types of intervals, which can be grouped two by two (by complementarity: each two intervals are the reverse of the other and add up to the tenth (interval sum of 16 eka) in eight categories that will be to be analysed individually:

1 eka - 15 eka
2 eka - 14 eka
3 eka - 13 eka
4 eka - 12 eka
5 eka - 11 eka
6 eka - 10 eka
7 eka - 9 eka
8 eka

==The intervals of 1 eka and 15 eka== 

The interval of one eka, the degree of the chromatic scale of Armodue equal to 3/4 of a semitone (75 cents), is the smallest interval of the system and is very close to the chromatic semitone postulated by Zarlino in his natural scale (based on simple ratios) and accounts for 70 cents.

This property of the eka makes it particularly euphonious and familiar to the ear: the eka is perceived as a natural interval not less than a semitone of the dodecatonic scale. In a free melodic improvisation, chromatic successions of consecutive ekas sound much like chromatic successions of semitones. Therefore, all the harmonic techniques inherent in chromaticism XXX can be applied in Armodue considering the eka as equivalent to a tempered semitone. The complement of one eka is the interval of 15 eka, comparable to a slightly enlarged major seventh of the dodecatonic system. The small size of the eka also makes it appropriate to evoke oriental sounds and atmospheres. The small intervals of 1 eka, 2 eka and 3 eka in Armodue lend themselves magnificently to the design of melodies and scales of exquisite modal and arabic flavour.

In Armodue the intervals of 1 and 15 eka eka are considered harsh dissonances and as such should be used with caution in chords. However, all rules may be applied that already govern the treatment of harsh dissonances in the dodecatonic system.

==The intervals of 2 eka and 14 eka== 

The interval of 2 eka corresponds to 3/2 of a tempered semitone (quantifies in 150 cents) and is found exactly between the eleventh and twelfth harmonic ov ther overtone series. It is the interval that is obtained by dividing the tenth of Armodue (the tempered octave) into eight equal parts, to form the [[8edo|8-equal tempered]] scale. Since only harmonics of higher number come close to it, it sounds particularly unnatural to the ear. This makes it suitable for geometric constructions where symmetry and artificialness prevail, for example in fractal sound structures.

In the dodecatonic system, the octave (1200 cents) divides up into an augmented fourth and a diminished fifth (600 cents), the tritone thus obtained can be divided into two minor thirds (300 cents), but the minor third may not further be subdivided into two parts. It is here that, where the possibilities of the dodecatonic system end, the possibilities of Armodue start, and we can continue in progressive subdivisions: the minor third, redefined as four eka (300 cents), is two eka plus two eka ( 150 cents); in turn, two eka is made up of two intervals of one eka each (75 cents). These algebraic/geometric properties of the initially considered interval of two eka make it particularly suitable for symmetric harmonic constructions - hence "speculative harmony". In practice, you can build speculative chords using only intervals of 16, 8, 4 or 2 eka between a voice and the adjacent one. One or more of notes thus obtained can later be altered one eka up or down; in this way, a harmonic construction that is rigidly squared first gains a new harmonic coloration particularly significant in the context.

The octatonic scale obtained by stepping through intervals of two consecutive eka has a very peculiar sound: from its lack of gravitation, it could be compared to the wholetone scale so much used by Debussy, but the interval of two eka is perceived by the western ear as closer to a semitone than to a tempered wholetone. The end result could be described as a scale about halfway between the chromatic and and the wholetone scale in the dodecatonic system. It can be very effective to use the 8-equal scale in chords and other types of scales in the melody, or vice versa: the introduction of different scale systems where the octatonic scale is primarily used adds variety to the harmonic-melodic architecture. Finally, it can be very fruitful in the composition with Armodue to recall that you have two types of "semitones" at your disposal: a "small semitone" (one eka) and a "big semitone" (two eka), next to the "wholetone" of Armodue (three eka). This in fact allows variations and potential completely absent in the dodecatonic system, where there is one - and only one - kind of semitone. For example, you can replace in a composition all intervals of one eka by intervals of two eka and vice versa, varying the type of "semitone small or big" perceived. A melody could fluctuate continually varying intervals in one and two eka, according to the principle of microvariation or according to techniques of musical minimalism (stubborn repetitions with small changes introduced). One very interesting thing a composer must keep in mind is the ambivalence of the 2 eka interval: this interval is exactly halfway between the wholetone and the tempered semitone.

This mediation between tone and semitone that is realized in Armodue can be much exploited in compositional technique, the principle can be to blur the contours of the "tone" (3 eka) and "semitone" (1 eka) replacing them both with the neutral and ambiguous interval 2 eka. Or, in the opposite direction, you can initially enunciate sound agglomerations where there are intervals of 2 eka, which are then replaced - "colouring" the harmonies and melodies - with intervals of 1 and 3 eka (just at will of the composer).

The interval of 2 eka and its complement of 14 eka are defined as neutral dissonances of Armodue.

==The intervals of 3 eka and 13 eka== 

The interval of 3 eka corresponds to the "wholetone" of Armodue (it is slightly wider than the tempered wholetone). This interval is particularly pleasing to the ear because it is very close to the natural tone that is formed with the seventh and the eighth harmonic (the tempered wholetone, by comparison, sounds less natural to the ear because it is formed with higher harmonics: the ninth and tenth). If you build scales using successions of the "wholetone" of Armodue, or proceeding for jumps of 3 eka, you get particularly evocative sounds - of vague pentatonic flavor.

The complement of the interval of 3 eka is the interval of 13 eka, which has a huge importance in Armodue as it corresponds to the natural minor seventh - the interval given by the ratio of the fourth harmonic with the seventh harmonic. Who has delved deeper into harmony topics is aware of how much a dominant seventh chord (example: C-E-G-Bb) played in 12-tone equal temperament differs from the corresponding natural chord found by overlapping the fourth, the fifth, the sixth and the seventh harmonic. This is due mainly to the non-negligible difference in pitch of the tempered minor seventh and the seventh harmonic (in this example the Bb). In Armodue, the minor seventh is returned to its natural pitch, the interval of 13 eka is perceived as very natural and euphonious.

From a philosophical point of view, with the two discussed intervals Armodue performs squaring the circle; within a "square", rigidly geometric structure (the division into sixteen notes), it places two "round", vague and exotic intervals as the natural wholetone and the natural minor seventh (in 3 and 13 eka). This latter consideration may prove as a very interesting opportunity for a composer.

The intervals of 3 and 13 eka are among the most suggestive intervals in Armodue and should be classified as sweet dissonances, as the tempered major second and minor seventh.

==The intervals of 4 eka and 12 eka== 

XXX

=Chapter 3: Creating scales with Armodue: modal systems= 

XXX

=Chapter 4: "Geometric" harmonic constructions with Armodue= 

XXX

=Chapter 5: "elastic" chords= 

XXX

Original HTML content:

<html><head><title>Armodue armonia</title></head><body><!-- ws:start:WikiTextTocRule:26:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --><div style="margin-left: 1em;"><a href="#Armodue: basic elements of harmony">Armodue: basic elements of harmony</a></div>
<!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --><div style="margin-left: 1em;"><a href="#Chapter 1: Two theses supporting the system">Chapter 1: Two theses supporting the system</a></div>
<!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --><div style="margin-left: 2em;"><a href="#Chapter 1: Two theses supporting the system-The supremacy of the fifth and and the seventh harmonic in Armodue">The supremacy of the fifth and and the seventh harmonic in Armodue</a></div>
<!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --><div style="margin-left: 2em;"><a href="#Chapter 1: Two theses supporting the system-The triple mean of the double diagonal / side of the square">The triple mean of the double diagonal / side of the square</a></div>
<!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --><div style="margin-left: 1em;"><a href="#Chapter 2: The interval table">Chapter 2: The interval table</a></div>
<!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --><div style="margin-left: 2em;"><a href="#Chapter 2: The interval table-Qualitative categories of intervals">Qualitative categories of intervals</a></div>
<!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --><div style="margin-left: 2em;"><a href="#Chapter 2: The interval table-The intervals of 1 eka and 15 eka">The intervals of 1 eka and 15 eka</a></div>
<!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --><div style="margin-left: 2em;"><a href="#Chapter 2: The interval table-The intervals of 2 eka and 14 eka">The intervals of 2 eka and 14 eka</a></div>
<!-- ws:end:WikiTextTocRule:34 --><!-- ws:start:WikiTextTocRule:35: --><div style="margin-left: 2em;"><a href="#Chapter 2: The interval table-The intervals of 3 eka and 13 eka">The intervals of 3 eka and 13 eka</a></div>
<!-- ws:end:WikiTextTocRule:35 --><!-- ws:start:WikiTextTocRule:36: --><div style="margin-left: 2em;"><a href="#Chapter 2: The interval table-The intervals of 4 eka and 12 eka">The intervals of 4 eka and 12 eka</a></div>
<!-- ws:end:WikiTextTocRule:36 --><!-- ws:start:WikiTextTocRule:37: --><div style="margin-left: 1em;"><a href="#Chapter 3: Creating scales with Armodue: modal systems">Chapter 3: Creating scales with Armodue: modal systems</a></div>
<!-- ws:end:WikiTextTocRule:37 --><!-- ws:start:WikiTextTocRule:38: --><div style="margin-left: 1em;"><a href="#Chapter 4: &quot;Geometric&quot; harmonic constructions with Armodue">Chapter 4: &quot;Geometric&quot; harmonic constructions with Armodue</a></div>
<!-- ws:end:WikiTextTocRule:38 --><!-- ws:start:WikiTextTocRule:39: --><div style="margin-left: 1em;"><a href="#Chapter 5: &quot;elastic&quot; chords">Chapter 5: &quot;elastic&quot; chords</a></div>
<!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --></div>
<!-- ws:end:WikiTextTocRule:40 --><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Armodue: basic elements of harmony"></a><!-- ws:end:WikiTextHeadingRule:0 --><strong>Armodue: basic elements of harmony</strong></h1>
 <br />
This is a translation of an article by Luca Attanasio. Original page in italian: <a class="wiki_link_ext" href="http://www.armodue.com/armonia.htm" rel="nofollow">http://www.armodue.com/armonia.htm</a><br />
<em>Note: This is a preliminary tranlsation. Parts that are still &quot;under construction&quot; are marked with &quot;XXX&quot;.</em><br />
<br />
<br />
<hr />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Chapter 1: Two theses supporting the system"></a><!-- ws:end:WikiTextHeadingRule:2 -->Chapter 1: Two theses supporting the system</h1>
 <br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Chapter 1: Two theses supporting the system-The supremacy of the fifth and and the seventh harmonic in Armodue"></a><!-- ws:end:WikiTextHeadingRule:4 -->The supremacy of the fifth and and the seventh harmonic in Armodue</h2>
 <br />
The twelve note system that has been ruling for several centuries is based on the third harmonic of the overtone series, which forms a perfect twelfth (octave-reducible to a perfect fifth) with the first harmonic or fundamental. Also, the pythagorean tradion - the cicle of fifths - is based on the perfect fifth and hence on the same frequency ratio 3:2.<br />
<br />
But, if in the twelve note system the pitch of the third harmonic - hence the perfect fourth and the perfect fifth - are almost perfectly respected (the tempered fourth and fifth differ only of the fiftieth part of a semitone from the natural fifth and fourth), this cannot be said about the odd harmonics (even harmonics are not counted because they are simply duplicates in the octave of odd harmonics) immediately above the third one: the fifth and seventh harmonic.<br />
<br />
In the tempered system the fifth and the seventh harmonic appear as major third and minor seventh intervals formed to the fundamental (equating all intervals to their octaver-reduced form for simplicity), but the sizes of the tempered major third and minor seventh do not match the sizes of the respective natural intervals.<br />
<br />
In Armodue, in contrast, intervals corresponding to those formed by the fifth and seventh harmonic are rendered with greater fidelity of intonation. In this sense, Armodue increases the consonance of the higher harmonics; in particular, it renders the pitch of the seventh harmonic at maximum naturalness.<br />
<br />
For this reason, especially important in Armodue are the interval of five eka (corresponding to the interval ratio given by the fifth harmonic) and the interval of 13 eka (corresponding to the interval ratio subsisting with the seventh harmonic). The circle of fifths which is the base of the dodecatonic system is replaced in Armodue by the cycle of 5 eka and the cycle of 13 eka, emphasizing the priority of the fifth and the seventh harmonic.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="Chapter 1: Two theses supporting the system-The triple mean of the double diagonal / side of the square"></a><!-- ws:end:WikiTextHeadingRule:6 -->The triple mean of the double diagonal / side of the square</h2>
 <br />
From a philosophical point of view, the system of twelve notes was justified in the past by this mathematical property: the arithmetic mean and the harmonic mean of the octave (interval ratio 2:1) correspond to the perfect fifth (ratio 3:2) and the perfect fourth (ratio 4:3), while the geometric mean divides the octave exatcly into two tritone intervals (ratio: square root of 2).<br />
<br />
Analogously, the philosophical foundation of Armodue and esadecafonia can be shown by calculating the three means of frequency geometrically equivalent to the ratio between the double of the diagonal (square root of 2 multiplied by 2) and the side length (of measure: 1) of a square.<br />
<br />
The arithmetic mean is exactly the interval of nine eka, the geometric mean exactly twelve eka and finally the harmonic mean exactly equal to fifteen eka.<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Chapter 2: The interval table"></a><!-- ws:end:WikiTextHeadingRule:8 -->Chapter 2: The interval table</h1>
 <br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Chapter 2: The interval table-Qualitative categories of intervals"></a><!-- ws:end:WikiTextHeadingRule:10 -->Qualitative categories of intervals</h2>
 <br />
Armodue consists of sixteen types of intervals, which can be grouped two by two (by complementarity: each two intervals are the reverse of the other and add up to the tenth (interval sum of 16 eka) in eight categories that will be to be analysed individually:<br />
<br />
1 eka - 15 eka<br />
2 eka - 14 eka<br />
3 eka - 13 eka<br />
4 eka - 12 eka<br />
5 eka - 11 eka<br />
6 eka - 10 eka<br />
7 eka - 9 eka<br />
8 eka<br />
<br />
<!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="Chapter 2: The interval table-The intervals of 1 eka and 15 eka"></a><!-- ws:end:WikiTextHeadingRule:12 -->The intervals of 1 eka and 15 eka</h2>
 <br />
The interval of one eka, the degree of the chromatic scale of Armodue equal to 3/4 of a semitone (75 cents), is the smallest interval of the system and is very close to the chromatic semitone postulated by Zarlino in his natural scale (based on simple ratios) and accounts for 70 cents.<br />
<br />
This property of the eka makes it particularly euphonious and familiar to the ear: the eka is perceived as a natural interval not less than a semitone of the dodecatonic scale. In a free melodic improvisation, chromatic successions of consecutive ekas sound much like chromatic successions of semitones. Therefore, all the harmonic techniques inherent in chromaticism XXX can be applied in Armodue considering the eka as equivalent to a tempered semitone. The complement of one eka is the interval of 15 eka, comparable to a slightly enlarged major seventh of the dodecatonic system. The small size of the eka also makes it appropriate to evoke oriental sounds and atmospheres. The small intervals of 1 eka, 2 eka and 3 eka in Armodue lend themselves magnificently to the design of melodies and scales of exquisite modal and arabic flavour.<br />
<br />
In Armodue the intervals of 1 and 15 eka eka are considered harsh dissonances and as such should be used with caution in chords. However, all rules may be applied that already govern the treatment of harsh dissonances in the dodecatonic system.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc7"><a name="Chapter 2: The interval table-The intervals of 2 eka and 14 eka"></a><!-- ws:end:WikiTextHeadingRule:14 -->The intervals of 2 eka and 14 eka</h2>
 <br />
The interval of 2 eka corresponds to 3/2 of a tempered semitone (quantifies in 150 cents) and is found exactly between the eleventh and twelfth harmonic ov ther overtone series. It is the interval that is obtained by dividing the tenth of Armodue (the tempered octave) into eight equal parts, to form the <a class="wiki_link" href="/8edo">8-equal tempered</a> scale. Since only harmonics of higher number come close to it, it sounds particularly unnatural to the ear. This makes it suitable for geometric constructions where symmetry and artificialness prevail, for example in fractal sound structures.<br />
<br />
In the dodecatonic system, the octave (1200 cents) divides up into an augmented fourth and a diminished fifth (600 cents), the tritone thus obtained can be divided into two minor thirds (300 cents), but the minor third may not further be subdivided into two parts. It is here that, where the possibilities of the dodecatonic system end, the possibilities of Armodue start, and we can continue in progressive subdivisions: the minor third, redefined as four eka (300 cents), is two eka plus two eka ( 150 cents); in turn, two eka is made up of two intervals of one eka each (75 cents). These algebraic/geometric properties of the initially considered interval of two eka make it particularly suitable for symmetric harmonic constructions - hence &quot;speculative harmony&quot;. In practice, you can build speculative chords using only intervals of 16, 8, 4 or 2 eka between a voice and the adjacent one. One or more of notes thus obtained can later be altered one eka up or down; in this way, a harmonic construction that is rigidly squared first gains a new harmonic coloration particularly significant in the context.<br />
<br />
The octatonic scale obtained by stepping through intervals of two consecutive eka has a very peculiar sound: from its lack of gravitation, it could be compared to the wholetone scale so much used by Debussy, but the interval of two eka is perceived by the western ear as closer to a semitone than to a tempered wholetone. The end result could be described as a scale about halfway between the chromatic and and the wholetone scale in the dodecatonic system. It can be very effective to use the 8-equal scale in chords and other types of scales in the melody, or vice versa: the introduction of different scale systems where the octatonic scale is primarily used adds variety to the harmonic-melodic architecture. Finally, it can be very fruitful in the composition with Armodue to recall that you have two types of &quot;semitones&quot; at your disposal: a &quot;small semitone&quot; (one eka) and a &quot;big semitone&quot; (two eka), next to the &quot;wholetone&quot; of Armodue (three eka). This in fact allows variations and potential completely absent in the dodecatonic system, where there is one - and only one - kind of semitone. For example, you can replace in a composition all intervals of one eka by intervals of two eka and vice versa, varying the type of &quot;semitone small or big&quot; perceived. A melody could fluctuate continually varying intervals in one and two eka, according to the principle of microvariation or according to techniques of musical minimalism (stubborn repetitions with small changes introduced). One very interesting thing a composer must keep in mind is the ambivalence of the 2 eka interval: this interval is exactly halfway between the wholetone and the tempered semitone.<br />
<br />
This mediation between tone and semitone that is realized in Armodue can be much exploited in compositional technique, the principle can be to blur the contours of the &quot;tone&quot; (3 eka) and &quot;semitone&quot; (1 eka) replacing them both with the neutral and ambiguous interval 2 eka. Or, in the opposite direction, you can initially enunciate sound agglomerations where there are intervals of 2 eka, which are then replaced - &quot;colouring&quot; the harmonies and melodies - with intervals of 1 and 3 eka (just at will of the composer).<br />
<br />
The interval of 2 eka and its complement of 14 eka are defined as neutral dissonances of Armodue.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:16:&lt;h2&gt; --><h2 id="toc8"><a name="Chapter 2: The interval table-The intervals of 3 eka and 13 eka"></a><!-- ws:end:WikiTextHeadingRule:16 -->The intervals of 3 eka and 13 eka</h2>
 <br />
The interval of 3 eka corresponds to the &quot;wholetone&quot; of Armodue (it is slightly wider than the tempered wholetone). This interval is particularly pleasing to the ear because it is very close to the natural tone that is formed with the seventh and the eighth harmonic (the tempered wholetone, by comparison, sounds less natural to the ear because it is formed with higher harmonics: the ninth and tenth). If you build scales using successions of the &quot;wholetone&quot; of Armodue, or proceeding for jumps of 3 eka, you get particularly evocative sounds - of vague pentatonic flavor.<br />
<br />
The complement of the interval of 3 eka is the interval of 13 eka, which has a huge importance in Armodue as it corresponds to the natural minor seventh - the interval given by the ratio of the fourth harmonic with the seventh harmonic. Who has delved deeper into harmony topics is aware of how much a dominant seventh chord (example: C-E-G-Bb) played in 12-tone equal temperament differs from the corresponding natural chord found by overlapping the fourth, the fifth, the sixth and the seventh harmonic. This is due mainly to the non-negligible difference in pitch of the tempered minor seventh and the seventh harmonic (in this example the Bb). In Armodue, the minor seventh is returned to its natural pitch, the interval of 13 eka is perceived as very natural and euphonious.<br />
<br />
From a philosophical point of view, with the two discussed intervals Armodue performs squaring the circle; within a &quot;square&quot;, rigidly geometric structure (the division into sixteen notes), it places two &quot;round&quot;, vague and exotic intervals as the natural wholetone and the natural minor seventh (in 3 and 13 eka). This latter consideration may prove as a very interesting opportunity for a composer.<br />
<br />
The intervals of 3 and 13 eka are among the most suggestive intervals in Armodue and should be classified as sweet dissonances, as the tempered major second and minor seventh.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name="Chapter 2: The interval table-The intervals of 4 eka and 12 eka"></a><!-- ws:end:WikiTextHeadingRule:18 -->The intervals of 4 eka and 12 eka</h2>
 <br />
XXX<br />
<br />
<!-- ws:start:WikiTextHeadingRule:20:&lt;h1&gt; --><h1 id="toc10"><a name="Chapter 3: Creating scales with Armodue: modal systems"></a><!-- ws:end:WikiTextHeadingRule:20 -->Chapter 3: Creating scales with Armodue: modal systems</h1>
 <br />
XXX<br />
<br />
<!-- ws:start:WikiTextHeadingRule:22:&lt;h1&gt; --><h1 id="toc11"><a name="Chapter 4: &quot;Geometric&quot; harmonic constructions with Armodue"></a><!-- ws:end:WikiTextHeadingRule:22 -->Chapter 4: &quot;Geometric&quot; harmonic constructions with Armodue</h1>
 <br />
XXX<br />
<br />
<!-- ws:start:WikiTextHeadingRule:24:&lt;h1&gt; --><h1 id="toc12"><a name="Chapter 5: &quot;elastic&quot; chords"></a><!-- ws:end:WikiTextHeadingRule:24 -->Chapter 5: &quot;elastic&quot; chords</h1>
 <br />
XXX</body></html>